Finite Group Databasebeta

← #366806

Subgroups — #366806

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C2×#1900C211yesmaximal, characteristic
3#20973C211yesmaximal, characteristic
4C2×#1855C211yesmaximal, characteristic, complemented
5#20700C211yesmaximal, characteristic, complemented
6#20389C211yesmaximal, characteristic, complemented
7C22×#343C211yesmaximal, characteristic, complemented
8#11365C212yesmaximal, complemented
9#11127C212yesmaximal, complemented
10#10943C212yesmaximal, complemented
11#10870C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#480—24no—
17C2×#480—21no—
18C2×#480—21no—
19C2×#480—22no—
20C2×#480C418yescomplemented
21C22×#108C2212yes—
22C22×#107C2212yes—
23C2×#418C2212yes—
24C22×#102C2212yes—
25C22×#100C2211yescharacteristic
26C22×#100C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C2×#405C2212yes—
30#2280C2211yescharacteristic
31#1900—24no—
32#1895—22no—
33#1893—22no—
34#1855—24no—
35#1824—22no—
36#1823—22no—
37#1753—22no—
38#1752—21no—
39C8×C22⋊C4—21no—
40C2×#343—24no—
41C2×#343C2211yescharacteristic
42C2×#343C2212yes—
43C2×#343C2211yescharacteristic
44C2×#343C2212yes—
45C2×#341C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48#1736C2212yes—
49#1729C2212yes—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C22×M16—22no—
56C22×M16—21no—
57C22×M16—22no—
58C22×M16—24no—
59C22×M16—24no—
60C22×M16—22no—
61C22×M16—42no—
62C22×M16—42no—
63C22×M16D414yes—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2312yes—
68C22×M16C2311yescharacteristic
69C23×C8—21no—
70C23×C8—21no—
71C23×C8C2312yes—
72C2×#121—22no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121—22no—
76C2×#121—21no—
77C2×#121—21no—
78C2×#121—22no—
79C2×#121C2×C414yes—
80C2×#117C2312yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C414yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93#480—28no—
94#480—28no—
95#480—216no—
96C2×#108—24no—
97C2×#108—24no—
98C2×#108—42no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107—24no—
102C2×#107C2×C418yes—
103#418—24no—
104#418—42no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102C2×C418yes—
110C2×#100—22no—
111C2×#100—22no—
112C2×#100—24no—
113C2×#100—42no—
114#407—22no—
115#407—22no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8C2×C414yes—
119C2×C22⋊C8D414yes—
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122C2×C22⋊C8D412yes—
123#405—24no—
124#405—42no—
125C2×#97—22no—
126C2×C4×C8—22no—
127#381—21no—
128#380D412yes—
129#380D412yes—
130#379—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133#343—22no—
134#343—22no—
135#343—24no—
136#343—24no—
137#341—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#336—22no—
142#336—22no—
143C25—21no—
144C25#4111yescharacteristic
145C22×D4—24no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—42no—
158C22×D4—84no—
159C22×D4C22×C412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—42no—
170C23×C4—84no—
171C23×C4C22×C412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4#4111yescharacteristic
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C2×M16—21no—
182C2×M16—22no—
183C2×M16—24no—
184C2×M16—21no—
185C2×M16—22no—
186C2×M16—28no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—28no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16—44no—
205C2×M16—48no—
206C2×M16—48no—
207C2×M16—44no—
208C2×M16—42no—
209C2×M16C2×D412yes—
210C2×M16C2×D412yes—
211C2×M16#4114yes—
212C2×M16#4114yes—
213C22×C8—21no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—24no—
219C22×C8—21no—
220C22×C8—22no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8—42no—
226C22×C8C22×C414yes—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8C2×D414yes—
230#121—24no—
231#121—24no—
232#121—24no—
233#121—24no—
234#121—42no—
235#121—42no—
236#121—42no—
237#121—42no—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C2×D414yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—42no—
274C2×C42—22no—
275#108—48no—
276#107—28no—
277#107—28no—
278#102—28no—
279#102—28no—
280#102—48no—
281#100—44no—
282#100—44no—
283C22⋊C8—42no—
284C22⋊C8—42no—
285C22⋊C8C22⋊C414yes—
286C22⋊C8C22⋊C414yes—
287C22⋊C8C22⋊C418yes—
288#97—44no—
289C4×C8—44no—
290#95—22no—
291#95—22no—
292C24—22no—
293C24—22no—
294C24—24no—
295C24—22no—
296C24—42no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—82no—
302C24—82no—
303C24C4×D412yes—
304C24#12111yescharacteristic
305C24#12111yescharacteristic
306C24#12412yes—
307C24C2×#4111yescharacteristic
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—88no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—88no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—88no—
365C22×C4—84no—
366C22×C4—82no—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12412yes—
377C22×C4#12712yes—
378C22×C4#12712yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389M16—22no—
390M16—24no—
391M16—22no—
392M16—24no—
393M16—22no—
394M16—24no—
395M16—48no—
396M16—48no—
397M16—42no—
398M16—42no—
399M16—42no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—42no—
405M16—42no—
406M16#12314yes—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—48no—
420C2×C8—48no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8C2×C22⋊C414yes—
429C2×C8C2×C22⋊C414yes—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—41no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—82no—
448C42—42no—
449C42—42no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—42no—
463C23—42no—
464C23—48no—
465C23—48no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—164no—
478C23—164no—
479C23#39012yes—
480C23#39012yes—
481C23#47214yes—
482C23C2×C4×D411yescharacteristic
483C23C2×#12111yescharacteristic
484C23C2×#12411yescharacteristic
485C23#52912yes—
486C23#53912yes—
487D4—44no—
488D4—44no—
489D4—88no—
490D4—88no—
491D4—81no—
492D4—81no—
493D4—81no—
494D4—81no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—84no—
518C2×C4—88no—
519C2×C4—82no—
520C2×C4—88no—
521C2×C4—84no—
522C2×C4—82no—
523C2×C4—84no—
524C2×C4—82no—
525C2×C4#39012yes—
526C2×C4#39012yes—
527C2×C4C22×C22⋊C411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C4×#4112yes—
530C2×C4#52012yes—
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12311yescharacteristic
533C2×C4#52511yescharacteristic
534C2×C4C2×#12711yescharacteristic
535C2×C4#53411yescharacteristic
536C2×C4#53611yescharacteristic
537C2×C4#54612yes—
538C2×C4#54612yes—
539C2×C4#54612yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4#54712yes—
543C2×C4#55112yes—
544C2×C4#55212yes—
545C8—24no—
546C8—24no—
547C8—24no—
548C8—42no—
549C8—44no—
550C8—81no—
551C8—81no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—24no—
557C22—24no—
558C22—28no—
559C22—21no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—44no—
565C22—48no—
566C22—44no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—162no—
572C22—162no—
573C22—164no—
574C22—164no—
575C22#197814yes—
576C22C2×#39011yescharacteristic
577C22C4×#12311yescharacteristic
578C22#232211yescharacteristic
579C22#242512yes—
580C22#248812yes—
581C22C2×#47212yes—
582C22#306912yes—
583C4—22no—
584C4—22no—
585C4—84no—
586C4—84no—
587C4—82no—
588C4#233912yes—
589C4#235912yes—
590C4#241211yescharacteristic
591C4#241311yescharacteristic
592C4#250111yescharacteristic
593C4#253911yescharacteristic
594C2—21no—
595C2—22no—
596C2—21no—
597C2—24no—
598C2—42no—
599C2—162no—
600C2#1399611yescharacteristic
601C2C2×#197811yescharacteristic
602C2#2125011yescharacteristic
603C2#2526414yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#1855C2semidirect
5. #20700C2semidirect
6. #20389C2semidirect
7. C22×#343C2semidirect
8. #11365C2semidirect
9. #11127C2semidirect
10. #10943C2semidirect
11. #10870C2semidirect
20. C2×#480C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#4802
C2×#19002
#209732
C2×#18552
#207002
#203892
C22×#3432
#113652
#111272
#109432
#108702